Question

An envelope contains eight bills: $2$ ones, $2$ fives, $2$ tens, and $2$ twenties. Two bills are drawn at random without replacement. What is the probability that their sum is $\$20$ or more?

$\dfrac{1}{4}$

$\dfrac{2}{7}$

$\dfrac{3}{7}$

$\dfrac{1}{2}$

$\dfrac{2}{3}$

Answer: D

Concepts:basic probability combinations casework

Difficulty rating: 1500

Solution:

There are $\binom{8}{2} = 28$ equally likely pairs of bills.

The sum is $\$20$ or more in these cases: both twenties ( $1$ way), one twenty with one of the six smaller bills ( $2 \cdot 6 = 12$ ways), or both tens ( $1$ way).

That is $1 + 12 + 1 = 14$ favorable pairs, so the probability is $\dfrac{14}{28} = \dfrac12.$

Thus, the correct answer is D.

2005 AMC 12B Problem 11

Answer: D

Difficulty rating: 1500

Solution:

Problem 11 in Other Years